Integrand size = 26, antiderivative size = 349 \[ \int \frac {(e x)^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {(A b-a B) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/2}}+\frac {(A b-7 a B) e (e x)^{3/2}}{6 a b^2 \sqrt {a+b x^2}}-\frac {(A b-7 a B) e^2 \sqrt {e x} \sqrt {a+b x^2}}{2 a b^{5/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {(A b-7 a B) e^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 a^{3/4} b^{11/4} \sqrt {a+b x^2}}-\frac {(A b-7 a B) e^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 a^{3/4} b^{11/4} \sqrt {a+b x^2}} \]
1/3*(A*b-B*a)*(e*x)^(7/2)/a/b/e/(b*x^2+a)^(3/2)+1/6*(A*b-7*B*a)*e*(e*x)^(3 /2)/a/b^2/(b*x^2+a)^(1/2)-1/2*(A*b-7*B*a)*e^2*(e*x)^(1/2)*(b*x^2+a)^(1/2)/ a/b^(5/2)/(a^(1/2)+x*b^(1/2))+1/2*(A*b-7*B*a)*e^(5/2)*(cos(2*arctan(b^(1/4 )*(e*x)^(1/2)/a^(1/4)/e^(1/2)))^2)^(1/2)/cos(2*arctan(b^(1/4)*(e*x)^(1/2)/ a^(1/4)/e^(1/2)))*EllipticE(sin(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/ 2))),1/2*2^(1/2))*(a^(1/2)+x*b^(1/2))*((b*x^2+a)/(a^(1/2)+x*b^(1/2))^2)^(1 /2)/a^(3/4)/b^(11/4)/(b*x^2+a)^(1/2)-1/4*(A*b-7*B*a)*e^(5/2)*(cos(2*arctan (b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)))^2)^(1/2)/cos(2*arctan(b^(1/4)*(e*x) ^(1/2)/a^(1/4)/e^(1/2)))*EllipticF(sin(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4 )/e^(1/2))),1/2*2^(1/2))*(a^(1/2)+x*b^(1/2))*((b*x^2+a)/(a^(1/2)+x*b^(1/2) )^2)^(1/2)/a^(3/4)/b^(11/4)/(b*x^2+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.10 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.28 \[ \int \frac {(e x)^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {2 e (e x)^{3/2} \left (a \left (A b-7 a B-3 b B x^2\right )+(-A b+7 a B) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{2},\frac {7}{4},-\frac {b x^2}{a}\right )\right )}{3 a b^2 \left (a+b x^2\right )^{3/2}} \]
(-2*e*(e*x)^(3/2)*(a*(A*b - 7*a*B - 3*b*B*x^2) + (-(A*b) + 7*a*B)*(a + b*x ^2)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[3/4, 5/2, 7/4, -((b*x^2)/a)]))/( 3*a*b^2*(a + b*x^2)^(3/2))
Time = 0.43 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {362, 252, 266, 834, 27, 761, 1510}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(e x)^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 362 |
| \(\displaystyle \frac {(e x)^{7/2} (A b-a B)}{3 a b e \left (a+b x^2\right )^{3/2}}-\frac {(A b-7 a B) \int \frac {(e x)^{5/2}}{\left (b x^2+a\right )^{3/2}}dx}{6 a b}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {(e x)^{7/2} (A b-a B)}{3 a b e \left (a+b x^2\right )^{3/2}}-\frac {(A b-7 a B) \left (\frac {3 e^2 \int \frac {\sqrt {e x}}{\sqrt {b x^2+a}}dx}{2 b}-\frac {e (e x)^{3/2}}{b \sqrt {a+b x^2}}\right )}{6 a b}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {(e x)^{7/2} (A b-a B)}{3 a b e \left (a+b x^2\right )^{3/2}}-\frac {(A b-7 a B) \left (\frac {3 e \int \frac {e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{b}-\frac {e (e x)^{3/2}}{b \sqrt {a+b x^2}}\right )}{6 a b}\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle \frac {(e x)^{7/2} (A b-a B)}{3 a b e \left (a+b x^2\right )^{3/2}}-\frac {(A b-7 a B) \left (\frac {3 e \left (\frac {\sqrt {a} e \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}-\frac {\sqrt {a} e \int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {a} e \sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}\right )}{b}-\frac {e (e x)^{3/2}}{b \sqrt {a+b x^2}}\right )}{6 a b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(e x)^{7/2} (A b-a B)}{3 a b e \left (a+b x^2\right )^{3/2}}-\frac {(A b-7 a B) \left (\frac {3 e \left (\frac {\sqrt {a} e \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}-\frac {\int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}\right )}{b}-\frac {e (e x)^{3/2}}{b \sqrt {a+b x^2}}\right )}{6 a b}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {(e x)^{7/2} (A b-a B)}{3 a b e \left (a+b x^2\right )^{3/2}}-\frac {(A b-7 a B) \left (\frac {3 e \left (\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^2}}-\frac {\int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}\right )}{b}-\frac {e (e x)^{3/2}}{b \sqrt {a+b x^2}}\right )}{6 a b}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {(e x)^{7/2} (A b-a B)}{3 a b e \left (a+b x^2\right )^{3/2}}-\frac {(A b-7 a B) \left (\frac {3 e \left (\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^2}}-\frac {\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {e^2 \sqrt {e x} \sqrt {a+b x^2}}{\sqrt {a} e+\sqrt {b} e x}}{\sqrt {b}}\right )}{b}-\frac {e (e x)^{3/2}}{b \sqrt {a+b x^2}}\right )}{6 a b}\) |
((A*b - a*B)*(e*x)^(7/2))/(3*a*b*e*(a + b*x^2)^(3/2)) - ((A*b - 7*a*B)*(-( (e*(e*x)^(3/2))/(b*Sqrt[a + b*x^2])) + (3*e*(-((-((e^2*Sqrt[e*x]*Sqrt[a + b*x^2])/(Sqrt[a]*e + Sqrt[b]*e*x)) + (a^(1/4)*Sqrt[e]*(Sqrt[a]*e + Sqrt[b] *e*x)*Sqrt[(a*e^2 + b*e^2*x^2)/(Sqrt[a]*e + Sqrt[b]*e*x)^2]*EllipticE[2*Ar cTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(b^(1/4)*Sqrt[a + b*x^2 ]))/Sqrt[b]) + (a^(1/4)*Sqrt[e]*(Sqrt[a]*e + Sqrt[b]*e*x)*Sqrt[(a*e^2 + b* e^2*x^2)/(Sqrt[a]*e + Sqrt[b]*e*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[e*x ])/(a^(1/4)*Sqrt[e])], 1/2])/(2*b^(3/4)*Sqrt[a + b*x^2])))/b))/(6*a*b)
3.9.16.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e *(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1)) I nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(p + 1)]))
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S imp[1/q Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Time = 3.02 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.86
| method | result | size |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {e^{2} x \left (A b -B a \right ) \sqrt {b e \,x^{3}+a e x}}{3 b^{4} \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {e^{3} x^{2} \left (A b -3 B a \right )}{2 b^{2} a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {\left (\frac {B \,e^{3}}{b^{2}}-\frac {e^{3} \left (A b -3 B a \right )}{4 b^{2} a}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{e x \sqrt {b \,x^{2}+a}}\) | \(300\) |
| default | \(-\frac {\left (6 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,b^{2} x^{2}-3 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,b^{2} x^{2}-42 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b \,x^{2}+21 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b \,x^{2}+6 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b -3 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b -42 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3}+21 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3}-6 A \,b^{3} x^{4}+18 B a \,b^{2} x^{4}-2 a A \,b^{2} x^{2}+14 B \,a^{2} b \,x^{2}\right ) e^{2} \sqrt {e x}}{12 x \,b^{3} a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(767\) |
1/e/x*(e*x)^(1/2)/(b*x^2+a)^(1/2)*((b*x^2+a)*e*x)^(1/2)*(-1/3*e^2/b^4*x*(A *b-B*a)*(b*e*x^3+a*e*x)^(1/2)/(x^2+a/b)^2+1/2/b^2*e^3*x^2/a*(A*b-3*B*a)/(( x^2+a/b)*b*e*x)^(1/2)+(B*e^3/b^2-1/4/b^2/a*e^3*(A*b-3*B*a))*(-a*b)^(1/2)/b *((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2)*(-2*(x-(-a*b)^(1/2)/b)/(-a*b)^( 1/2)*b)^(1/2)*(-x/(-a*b)^(1/2)*b)^(1/2)/(b*e*x^3+a*e*x)^(1/2)*(-2*(-a*b)^( 1/2)/b*EllipticE(((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2),1/2*2^(1/2))+(- a*b)^(1/2)/b*EllipticF(((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2),1/2*2^(1/ 2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.51 \[ \int \frac {(e x)^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {3 \, {\left ({\left (7 \, B a b^{2} - A b^{3}\right )} e^{2} x^{4} + 2 \, {\left (7 \, B a^{2} b - A a b^{2}\right )} e^{2} x^{2} + {\left (7 \, B a^{3} - A a^{2} b\right )} e^{2}\right )} \sqrt {b e} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (3 \, {\left (3 \, B a b^{2} - A b^{3}\right )} e^{2} x^{3} + {\left (7 \, B a^{2} b - A a b^{2}\right )} e^{2} x\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{6 \, {\left (a b^{5} x^{4} + 2 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}} \]
-1/6*(3*((7*B*a*b^2 - A*b^3)*e^2*x^4 + 2*(7*B*a^2*b - A*a*b^2)*e^2*x^2 + ( 7*B*a^3 - A*a^2*b)*e^2)*sqrt(b*e)*weierstrassZeta(-4*a/b, 0, weierstrassPI nverse(-4*a/b, 0, x)) + (3*(3*B*a*b^2 - A*b^3)*e^2*x^3 + (7*B*a^2*b - A*a* b^2)*e^2*x)*sqrt(b*x^2 + a)*sqrt(e*x))/(a*b^5*x^4 + 2*a^2*b^4*x^2 + a^3*b^ 3)
Timed out. \[ \int \frac {(e x)^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(e x)^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(e x)^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(e x)^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^{5/2}}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \]